Determinants of adjacency matrices of graphs
We study the set of all determinants of adjacency matrices of graphs with a given number of vertices. Using Brendan McKay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2012-12-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/?_action=showPDF&article=2041&_ob=dfea7756845067696c4a750520b02fe7&fileName=full_text.pdf. |
| _version_ | 1819154199241293824 |
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| author | Alireza Abdollahi |
| author_facet | Alireza Abdollahi |
| author_sort | Alireza Abdollahi |
| collection | DOAJ |
| description | We study the set of all determinants of adjacency matrices of graphs with a given number of vertices. Using Brendan McKay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. Using an idea of M. Newman, it is proved that if $G$ is a graph with $n$ vertices and ${d_1,dots,d_n}$ is the set of vertex degrees of $G$, then $gcd(2m,d^2)$ divides the determinant of the adjacency matrix of $G$, where $d=gcd(d_1,dots,d_n)$. Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained. |
| first_indexed | 2024-12-22T15:17:17Z |
| format | Article |
| id | doaj.art-aa5e0b9f855e40808c469b4e403fc01f |
| institution | Directory Open Access Journal |
| issn | 2251-8657 2251-8665 |
| language | English |
| last_indexed | 2024-12-22T15:17:17Z |
| publishDate | 2012-12-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | Transactions on Combinatorics |
| spelling | doaj.art-aa5e0b9f855e40808c469b4e403fc01f2022-12-21T18:21:43ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652012-12-0114916Determinants of adjacency matrices of graphsAlireza AbdollahiWe study the set of all determinants of adjacency matrices of graphs with a given number of vertices. Using Brendan McKay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. Using an idea of M. Newman, it is proved that if $G$ is a graph with $n$ vertices and ${d_1,dots,d_n}$ is the set of vertex degrees of $G$, then $gcd(2m,d^2)$ divides the determinant of the adjacency matrix of $G$, where $d=gcd(d_1,dots,d_n)$. Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained.http://www.combinatorics.ir/?_action=showPDF&article=2041&_ob=dfea7756845067696c4a750520b02fe7&fileName=full_text.pdf.Determinantadjacency matrices of graphsmaximum determinant |
| spellingShingle | Alireza Abdollahi Determinants of adjacency matrices of graphs Transactions on Combinatorics Determinant adjacency matrices of graphs maximum determinant |
| title | Determinants of adjacency matrices of graphs |
| title_full | Determinants of adjacency matrices of graphs |
| title_fullStr | Determinants of adjacency matrices of graphs |
| title_full_unstemmed | Determinants of adjacency matrices of graphs |
| title_short | Determinants of adjacency matrices of graphs |
| title_sort | determinants of adjacency matrices of graphs |
| topic | Determinant adjacency matrices of graphs maximum determinant |
| url | http://www.combinatorics.ir/?_action=showPDF&article=2041&_ob=dfea7756845067696c4a750520b02fe7&fileName=full_text.pdf. |
| work_keys_str_mv | AT alirezaabdollahi determinantsofadjacencymatricesofgraphs |